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review derivation: particle in a 1D box

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
20 change variable X to Y
  1. 9988949211; locally 1231131:
    \((\sin(x))^2 = \frac{1 - \cos(2 x)}{2}\)
    \(\sin^{2}{\left(pdg_{1464} \right)} = \frac{1}{2} - \frac{\cos{\left(2 pdg_{1464} \right)}}{2}\)
  1. 0009484724:
    \(\frac{n \pi}{W}x\)
    \(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}}\)
  2. 0004934845:
    \(x\)
    \(pdg_{1464}\)
  1. 7575738420; locally 0100404:
    \(\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}\)
    \(\sin^{2}{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)} = \frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\)
LHS diff is sin(pdg1464)**2 - sin(pdg1592*pdg3141*pdg4037/pdg2523)**2 RHS diff is -cos(2*pdg1464)/2 + cos(2*pdg1464*pdg1592*pdg3141/pdg2523)/2 9988949211:
7575738420:
9988949211:
7575738420:
32 square root both sides
  1. 8485867742; locally 1029384:
    \(\frac{2}{W} = a^2\)
    \(\frac{2}{pdg_{2523}} = pdg_{9139}^{2}\)
  1. 9485747245; locally 9394857:
    \(\sqrt{\frac{2}{W}} = a\)
    \(\sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} = pdg_{9139}\)
  2. 9485747246; locally 9394858:
    \(-\sqrt{\frac{2}{W}} = a\)
    \(- \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} = pdg_{9139}\)
no check performed 8485867742:
9485747245:
9485747246:
8485867742:
9485747245:
9485747246:
2 declare guess solution
  1. 5727578862; locally 7572748:
    \(\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)\)
    \(pdg_{9199}\)
  1. 8582885111; locally 7572118:
    \(\psi(x) = a \sin(kx) + b \cos(kx)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{4037} \right)} = pdg_{1939} \cos{\left(pdg_{4037} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{4037} pdg_{5321} \right)}\)
no validation is available for declarations 5727578862:
8582885111:
5727578862:
8582885111:
33 substitute LHS of expr 1 into expr 2
  1. 9485747245; locally 9394857:
    \(\sqrt{\frac{2}{W}} = a\)
    \(\sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} = pdg_{9139}\)
  2. 2944838499; locally 3452131:
    \(\psi(x) = a \sin(\frac{n \pi}{W} x)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\)
  1. 9393939991; locally 8474766:
    \(\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = - \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}\)
LHS diff is 0 RHS diff is pdg9139*sin(pdg1592*pdg3141*pdg4037/pdg2523) + sqrt(2)*sqrt(1/pdg2523)*sin(pdg1464*pdg1592*pdg3141/pdg2523) 9485747245:
2944838499:
9393939991:
9485747245:
2944838499:
9393939991:
27 change variable X to Y
  1. 5857434758; locally 0021030:
    \(\int a dx = a x\)
    \(\int pdg_{9139}\, dpdg_{1464} = pdg_{1464} pdg_{9139}\)
  1. 0002929944:
    \(1/2\)
    \(\frac{1}{2}\)
  2. 0004948585:
    \(a\)
    \(pdg_{9139}\)
  1. 8575746378; locally 9339495:
    \(\int \frac{1}{2} dx = \frac{1}{2} x\)
    \(\int \frac{1}{2}\, dpdg_{1464} = \frac{pdg_{1464}}{2}\)
LHS diff is pdg1464*(pdg9139 - 1/2) RHS diff is pdg1464*(pdg9139 - 1/2) 5857434758:
8575746378:
5857434758:
8575746378:
38 simplify
  1. 8575748999; locally 2838288:
    \(\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right)\)
    \(\frac{d^{2} \left(pdg_{1939} \cos{\left(pdg_{1464} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{1464} pdg_{5321} \right)}\right)}{pdg_{9199}^{2}} = - pdg_{5321}^{2} \left(pdg_{1939} \cos{\left(pdg_{1464} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{1464} pdg_{5321} \right)}\right)\)
  1. 8485757728; locally 8474762:
    \(a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)\)
    \(pdg_{9199}\)
Nothing to split 8575748999:
8485757728:
8575748999:
8485757728:
26 declare identity
  1. 5857434758; locally 0021030:
    \(\int a dx = a x\)
    \(\int pdg_{9139}\, dpdg_{1464} = pdg_{1464} pdg_{9139}\)
no validation is available for declarations 5857434758:
5857434758:
24 declare identity
  1. 0948572140; locally 3992939:
    \(\int \cos(a x) dx = \frac{1}{a}\sin(a x)\)
    \(\int \cos{\left(pdg_{1464} pdg_{9139} \right)}\, dpdg_{9199} = \frac{\sin{\left(pdg_{1464} pdg_{9139} \right)}}{pdg_{9139}}\)
no validation is available for declarations 0948572140:
0948572140:
28 substitute LHS of expr 1 into expr 2
  1. 8575746378; locally 9339495:
    \(\int \frac{1}{2} dx = \frac{1}{2} x\)
    \(\int \frac{1}{2}\, dpdg_{1464} = \frac{pdg_{1464}}{2}\)
  2. 1202310110; locally 0203020:
    \(\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
    \(\frac{1}{pdg_{9139}^{2}} = \int\limits_{0}^{pdg_{2523}} \left(\frac{pdg_{9199}}{2} - \frac{\int\limits_{0}^{pdg_{2523}} \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037}}{2}\right)\, dpdg_{4037}\)
  1. 1202312210; locally 8584733:
    \(\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
    \(\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2} - \frac{\int\limits_{0}^{pdg_{2523}} \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037}}{2}\)
LHS diff is 0 RHS diff is Piecewise((pdg2523*(2*pdg1592*pdg3141*pdg9199 - 2*pdg1592*pdg3141 - pdg2523*sin(2*pdg1592*pdg3141) + sin(2*pdg1592*pdg3141))/(4*pdg1592*pdg3141), Ne(2*pdg1592*pdg3141/pdg2523, 0)), (pdg2523*(-pdg2523 + pdg9199)/2, True)) 8575746378:
1202310110:
1202312210:
8575746378:
1202310110:
1202312210:
34 substitute LHS of expr 1 into expr 2
  1. 9485747246; locally 9394858:
    \(-\sqrt{\frac{2}{W}} = a\)
    \(- \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} = pdg_{9139}\)
  2. 2944838499; locally 3452131:
    \(\psi(x) = a \sin(\frac{n \pi}{W} x)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\)
  1. 9393939992; locally 8474765:
    \(\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}\)
LHS diff is 0 RHS diff is pdg9139*sin(pdg1592*pdg3141*pdg4037/pdg2523) - sqrt(2)*sqrt(1/pdg2523)*sin(pdg1464*pdg1592*pdg3141/pdg2523) 9485747246:
2944838499:
9393939992:
9485747246:
2944838499:
9393939992:
23 expand integrand
  1. 9858028950; locally 0495054:
    \(\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx\)
    \(\frac{1}{pdg_{9139}^{2}} = \int\limits_{0}^{pdg_{2523}} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\right)\, dpdg_{1464}\)
  1. 1202310110; locally 0203020:
    \(\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
    \(\frac{1}{pdg_{9139}^{2}} = \int\limits_{0}^{pdg_{2523}} \left(\frac{pdg_{9199}}{2} - \frac{\int\limits_{0}^{pdg_{2523}} \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037}}{2}\right)\, dpdg_{4037}\)
no check performed 9858028950:
1202310110:
9858028950:
1202310110:
13 normalization condition
  1. 1934748140; locally 7575626:
    \(\int |\psi(x)|^2 dx = 1\)
    \(\int \left|{\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}}\right|^{2}\, dpdg_{9199} = 1\)
no validation is available for assumptions 1934748140:
1934748140:
17 substitute LHS of expr 1 into expr 2
  1. 2944838499; locally 3452131:
    \(\psi(x) = a \sin(\frac{n \pi}{W} x)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\)
  2. 4857472413; locally 0595847:
    \(1 = \int \psi(x)\psi(x)^* dx\)
    \(pdg_{9199}\)
  1. 0203024440; locally 0495950:
    \(1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx\)
    \(1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)} \overline{\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}}\, dpdg_{1464}\)
Nothing to split 2944838499:
4857472413:
0203024440:
2944838499:
4857472413:
0203024440:
25 change variable X to Y
  1. 0948572140; locally 3992939:
    \(\int \cos(a x) dx = \frac{1}{a}\sin(a x)\)
    \(\int \cos{\left(pdg_{1464} pdg_{9139} \right)}\, dpdg_{9199} = \frac{\sin{\left(pdg_{1464} pdg_{9139} \right)}}{pdg_{9139}}\)
  1. 0009485858:
    \(\frac{2n\pi}{W}\)
    \(\frac{2 pdg_{1592} pdg_{3141}}{pdg_{2523}}\)
  2. 0004831494:
    \(a\)
    \(pdg_{9139}\)
  1. 7564894985; locally 4948377:
    \(\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\)
    \(\int \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037} = \frac{pdg_{2523} \sin{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}}{2 pdg_{1592} pdg_{3141}}\)
LHS diff is pdg9199*cos(pdg1464*pdg9139) - Piecewise((pdg2523*sin(2*pdg1592*pdg3141*pdg4037/pdg2523)/(2*pdg1592*pdg3141), Ne(2*pdg1592*pdg3141/pdg2523, 0)), (pdg4037, True)) RHS diff is sin(pdg1464*pdg9139)/pdg9139 - pdg2523*sin(2*pdg1592*pdg3141*pdg4037/pdg2523)/(2*pdg1592*pdg3141) 0948572140:
7564894985:
0948572140:
7564894985:
15 swap LHS with RHS
  1. 1934748140; locally 7575626:
    \(\int |\psi(x)|^2 dx = 1\)
    \(\int \left|{\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}}\right|^{2}\, dpdg_{9199} = 1\)
  1. 8572657110; locally 5577567:
    \(1 = \int |\psi(x)|^2 dx\)
    \(1 = \int \left|{\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}}\right|^{2}\, dpdg_{1464}\)
LHS diff is pdg9199*Abs(pdg9489(pdg1464))**2 - Integral(Abs(pdg9489(pdg1464))**2, pdg1464) RHS diff is pdg9199*Abs(pdg9489(pdg1464))**2 - Integral(Abs(pdg9489(pdg1464))**2, pdg1464) 1934748140:
8572657110:
1934748140:
8572657110:
37 substitute RHS of expr 1 into expr 2
  1. 5727578862; locally 7572748:
    \(\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)\)
    \(pdg_{9199}\)
  2. 8582885111; locally 7572118:
    \(\psi(x) = a \sin(kx) + b \cos(kx)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{4037} \right)} = pdg_{1939} \cos{\left(pdg_{4037} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{4037} pdg_{5321} \right)}\)
  1. 8575748999; locally 2838288:
    \(\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right)\)
    \(\frac{d^{2} \left(pdg_{1939} \cos{\left(pdg_{1464} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{1464} pdg_{5321} \right)}\right)}{pdg_{9199}^{2}} = - pdg_{5321}^{2} \left(pdg_{1939} \cos{\left(pdg_{1464} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{1464} pdg_{5321} \right)}\right)\)
Nothing to split 5727578862:
8582885111:
8575748999:
5727578862:
8582885111:
8575748999:
18 substitute LHS of expr 1 into expr 2
  1. 8849289982; locally 3452132:
    \(\psi(x)^* = a \sin(\frac{n \pi}{W} x)\)
    \(\overline{\operatorname{pdg_{9489}}{\left(pdg_{4037} \right)}} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\)
  2. 0203024440; locally 0495950:
    \(1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx\)
    \(1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)} \overline{\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}}\, dpdg_{1464}\)
  1. 8889444440; locally 8478550:
    \(1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx\)
    \(1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139}^{2} \sin^{2}{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}\, dpdg_{1464}\)
LHS diff is 0 RHS diff is pdg9139*(-pdg9139*Piecewise((pdg2523*(pdg1592*pdg3141/2 - sin(pdg1592*pdg3141)*cos(pdg1592*pdg3141)/2)/(pdg1592*pdg3141), Ne(pdg1592*pdg3141/pdg2523, 0)), (0, True)) + Integral(sin(pdg1464*pdg1592*pdg3141/pdg2523)*conjugate(pdg9489(pdg1464)), (pdg1464, 0, pdg2523))) 8849289982:
0203024440:
8889444440:
8849289982:
0203024440:
8889444440:
14 conjugate function X
  1. 2944838499; locally 3452131:
    \(\psi(x) = a \sin(\frac{n \pi}{W} x)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\)
  1. 0009587738:
    \(\psi\)
    \(pdg_{9489}\)
  1. 8849289982; locally 3452132:
    \(\psi(x)^* = a \sin(\frac{n \pi}{W} x)\)
    \(\overline{\operatorname{pdg_{9489}}{\left(pdg_{4037} \right)}} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\)
no check performed 2944838499:
8849289982:
2944838499:
8849289982:
31 multiply both sides by
  1. 4857475848; locally 9493949:
    \(\frac{1}{a^2} = \frac{W}{2}\)
    \(\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2}\)
  1. 0009485857:
    \(a^2\frac{2}{W}\)
    \(\frac{2 pdg_{9139}^{2}}{pdg_{2523}}\)
  1. 8485867742; locally 1029384:
    \(\frac{2}{W} = a^2\)
    \(\frac{2}{pdg_{2523}} = pdg_{9139}^{2}\)
valid 4857475848:
8485867742:
4857475848:
8485867742:
22 divide both sides by
  1. 8576785890; locally 9485800:
    \(1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx\)
    \(1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139}^{2} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\right)\, dpdg_{1464}\)
  1. 0000040490:
    \(a^2\)
    \(pdg_{9139}^{2}\)
  1. 9858028950; locally 0495054:
    \(\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx\)
    \(\frac{1}{pdg_{9139}^{2}} = \int\limits_{0}^{pdg_{2523}} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\right)\, dpdg_{1464}\)
valid 8576785890:
9858028950:
8576785890:
9858028950:
40 claim LHS equals RHS
  1. 8484544728; locally 1214762:
    \(-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)\)
    \(pdg_{4037}\)
Nothing to split 8484544728:
8484544728:
16 expand magnitude to conjugate
  1. 8572657110; locally 5577567:
    \(1 = \int |\psi(x)|^2 dx\)
    \(1 = \int \left|{\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}}\right|^{2}\, dpdg_{1464}\)
  1. 0009458842:
    \(\psi(x)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}\)
  1. 4857472413; locally 0595847:
    \(1 = \int \psi(x)\psi(x)^* dx\)
    \(pdg_{9199}\)
Nothing to split 8572657110:
4857472413:
8572657110:
4857472413:
35 declare final expr
  1. 9393939992; locally 8474765:
    \(\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}\)
no validation is available for declarations 9393939992:
9393939992:
30 simplify
  1. 0439492440; locally 0405049:
    \(\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W\)
    \(\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2} - \frac{pdg_{2523} \sin{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}}{4 pdg_{1592} pdg_{3141}}\)
  1. 4857475848; locally 9493949:
    \(\frac{1}{a^2} = \frac{W}{2}\)
    \(\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2}\)
LHS diff is 0 RHS diff is -pdg2523*sin(2*pdg1592*pdg3141*pdg4037/pdg2523)/(4*pdg1592*pdg3141) 0439492440:
4857475848:
0439492440:
4857475848:
19 declare identity
  1. 9988949211; locally 1231131:
    \((\sin(x))^2 = \frac{1 - \cos(2 x)}{2}\)
    \(\sin^{2}{\left(pdg_{1464} \right)} = \frac{1}{2} - \frac{\cos{\left(2 pdg_{1464} \right)}}{2}\)
no validation is available for declarations 9988949211:
9988949211:
29 substitute RHS of expr 1 into expr 2
  1. 7564894985; locally 4948377:
    \(\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\)
    \(\int \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037} = \frac{pdg_{2523} \sin{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}}{2 pdg_{1592} pdg_{3141}}\)
  2. 1202312210; locally 8584733:
    \(\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
    \(\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2} - \frac{\int\limits_{0}^{pdg_{2523}} \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037}}{2}\)
  1. 0439492440; locally 0405049:
    \(\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W\)
    \(\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2} - \frac{pdg_{2523} \sin{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}}{4 pdg_{1592} pdg_{3141}}\)
LHS diff is 0 RHS diff is -Piecewise((pdg2523*sin(2*pdg1592*pdg3141)/(2*pdg1592*pdg3141), Ne(2*pdg1592*pdg3141/pdg2523, 0)), (pdg2523, True))/2 + pdg2523*sin(2*pdg1592*pdg3141*pdg4037/pdg2523)/(4*pdg1592*pdg3141) 7564894985:
1202312210:
0439492440:
7564894985:
1202312210:
0439492440:
5 LHS of expr 1 equals LHS of expr 2
  1. 9585727710; locally 8577781:
    \(\psi(x=0) = 0\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{1464} = 0 \right)} = 0\)
  2. 8582885111; locally 7572118:
    \(\psi(x) = a \sin(kx) + b \cos(kx)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{4037} \right)} = pdg_{1939} \cos{\left(pdg_{4037} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{4037} pdg_{5321} \right)}\)
  1. 8577275751; locally 7547581:
    \(0 = a \sin(0) + b\cos(0)\)
    \(0 = pdg_{1939}\)
input diff is -pdg9489(pdg4037) + pdg9489(Eq(pdg1464, 0)) diff is 0 diff is -pdg1939*cos(pdg4037*pdg5321) + pdg1939 - pdg9139*sin(pdg4037*pdg5321) 9585727710:
8582885111:
8577275751:
9585727710:
8582885111:
8577275751:
7 substitute RHS of expr 1 into expr 2
  1. 1293913110; locally 7572859:
    \(0 = b\)
    \(0 = pdg_{1939}\)
  2. 8582885111; locally 7572118:
    \(\psi(x) = a \sin(kx) + b \cos(kx)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{4037} \right)} = pdg_{1939} \cos{\left(pdg_{4037} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{4037} pdg_{5321} \right)}\)
  1. 9059289981; locally 7562671:
    \(\psi(x) = a \sin(k x)\)
    \(pdg_{1464}\)
Nothing to split 1293913110:
8582885111:
9059289981: no LHS/RHS split
1293913110:
8582885111:
9059289981: N/A
3 boundary condition for expr
  1. 5727578862; locally 7572748:
    \(\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)\)
    \(pdg_{9199}\)
  1. 9585727710; locally 8577781:
    \(\psi(x=0) = 0\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{1464} = 0 \right)} = 0\)
no validation is available for assumptions 5727578862:
9585727710:
5727578862:
9585727710:
4 boundary condition for expr
  1. 5727578862; locally 7572748:
    \(\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)\)
    \(pdg_{9199}\)
  1. 9495857278; locally 8585727:
    \(\psi(x=W) = 0\)
    \(pdg_{2523}\)
no validation is available for assumptions 5727578862:
9495857278: no LHS/RHS split
5727578862:
9495857278: N/A
10 expr 1 is true under condition expr 2
  1. 1020010291; locally 8577672:
    \(0 = a \sin(k W)\)
    \(0 = pdg_{9139} \sin{\left(pdg_{2523} pdg_{5321} \right)}\)
  2. 1857710291; locally 8577711:
    \(0 = a \sin(n \pi)\)
    \(0 = pdg_{9139} \sin{\left(pdg_{1592} pdg_{3141} \right)}\)
  1. 1010923823; locally 9847600:
    \(k W = n \pi\)
    \(pdg_{2523} pdg_{5321} = pdg_{1592} pdg_{3141}\)
no check performed 1020010291: error for dim with 1020010291
1857710291:
1010923823:
1020010291: N/A
1857710291:
1010923823:
9 declare identity
  1. 1857710291; locally 8577711:
    \(0 = a \sin(n \pi)\)
    \(0 = pdg_{9139} \sin{\left(pdg_{1592} pdg_{3141} \right)}\)
no validation is available for declarations 1857710291:
1857710291:
12 substitute RHS of expr 1 into expr 2
  1. 1858772113; locally 9495882:
    \(k = \frac{n \pi}{W}\)
    \(pdg_{5321} = \frac{pdg_{1592} pdg_{3141}}{pdg_{2523}}\)
  2. 9059289981; locally 7562671:
    \(\psi(x) = a \sin(k x)\)
    \(pdg_{1464}\)
  1. 2944838499; locally 3452131:
    \(\psi(x) = a \sin(\frac{n \pi}{W} x)\)
    \(\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\)
Nothing to split 1858772113:
9059289981: no LHS/RHS split
2944838499:
1858772113:
9059289981: N/A
2944838499:
6 simplify
  1. 8577275751; locally 7547581:
    \(0 = a \sin(0) + b\cos(0)\)
    \(0 = pdg_{1939}\)
  1. 1293913110; locally 7572859:
    \(0 = b\)
    \(0 = pdg_{1939}\)
valid 8577275751:
1293913110:
8577275751:
1293913110:
1 declare initial expr
  1. 5727578862; locally 7572748:
    \(\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)\)
    \(pdg_{9199}\)
no validation is available for declarations 5727578862:
5727578862:
21 substitute RHS of expr 1 into expr 2
  1. 7575738420; locally 0100404:
    \(\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}\)
    \(\sin^{2}{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)} = \frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\)
  2. 8889444440; locally 8478550:
    \(1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx\)
    \(1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139}^{2} \sin^{2}{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}\, dpdg_{1464}\)
  1. 8576785890; locally 9485800:
    \(1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx\)
    \(1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139}^{2} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\right)\, dpdg_{1464}\)
valid 7575738420:
8889444440:
8576785890:
7575738420:
8889444440:
8576785890:
11 divide both sides by
  1. 1010923823; locally 9847600:
    \(k W = n \pi\)
    \(pdg_{2523} pdg_{5321} = pdg_{1592} pdg_{3141}\)
  1. 0001334112:
    \(W\)
    \(pdg_{2523}\)
  1. 1858772113; locally 9495882:
    \(k = \frac{n \pi}{W}\)
    \(pdg_{5321} = \frac{pdg_{1592} pdg_{3141}}{pdg_{2523}}\)
valid 1010923823:
1858772113:
1010923823:
1858772113:
39 simplify
  1. 8485757728; locally 8474762:
    \(a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)\)
    \(pdg_{9199}\)
  1. 8484544728; locally 1214762:
    \(-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)\)
    \(pdg_{4037}\)
Nothing to split 8485757728:
8484544728:
8485757728:
8484544728:
8 LHS of expr 1 equals LHS of expr 2
  1. 9495857278; locally 8585727:
    \(\psi(x=W) = 0\)
    \(pdg_{2523}\)
  2. 9059289981; locally 7562671:
    \(\psi(x) = a \sin(k x)\)
    \(pdg_{1464}\)
  1. 1020010291; locally 8577672:
    \(0 = a \sin(k W)\)
    \(0 = pdg_{9139} \sin{\left(pdg_{2523} pdg_{5321} \right)}\)
Nothing to split 9495857278: no LHS/RHS split
9059289981: no LHS/RHS split
1020010291: error for dim with 1020010291
9495857278: N/A
9059289981: N/A
1020010291: N/A
Physics Derivation Graph: Steps for particle in a 1D box

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
1464 variable x
\(x\)
['real']
140
9489 variable \psi
\(\psi\)
complex dimensionless none
  • str_note
27
4037 variable x
\(x\)
['real']
  • length: 1
position 47
9139 variable a
\(a\)
['real']
45
2523 variable W
\(W\)
real
  • length: 1
width 25
9199 variable dx
\(dx\)
['real']
  • length: 1
15
1939 variable b
\(b\)
['real']
20
5321 variable k
\(k\)
['real']
  • length: -1
angular wavenumber 13
1592 variable n
\(n\)
integer
index 23
3141 constant \pi
\(\pi\)
['real']
pi 3.1415   dimensionless
55
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